Finite Group Databasebeta

← #25085989

Subgroups — #25085989

Order 11008.

736 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19146938C211yesmaximal, characteristic, complemented
2#19146402C211yesmaximal, characteristic, complemented
3#19146196C211yesmaximal, characteristic, complemented
4#19134267C211yesmaximal, characteristic, complemented
5#19134081C211yesmaximal, characteristic, complemented
6#19134034C211yesmaximal, characteristic, complemented
7#19133982C211yesmaximal, characteristic, complemented
8#19133936C211yesmaximal, characteristic, complemented
9#19133830C211yesmaximal, characteristic, complemented
10#19133828C211yesmaximal, characteristic, complemented
11#19133772C211yesmaximal, characteristic, complemented
12#19133684C211yesmaximal, characteristic, complemented
13#19133424C211yesmaximal, characteristic, complemented
14#19133419C211yesmaximal, characteristic, complemented
15C43×#3283C211yesmaximal, characteristic, complemented
16C2×#13063349—21no—
17#14583553C2211yescharacteristic
18D4×D172C2211yescharacteristic
19#14583302C2211yescharacteristic, complemented
20#14583302C2211yescharacteristic, complemented
21#14583272C2211yescharacteristic
22#14583247C2211yescharacteristic
23#14583247C2211yescharacteristic
24#14583212—21no—
25#14583212—21no—
26#14583210—21no—
27#14583179C2211yescharacteristic, complemented
28#14583176C2211yescharacteristic
29#14583176C2211yescharacteristic
30#14583146—21no—
31#14583133—21no—
32#14583133—21no—
33#14583104C2211yescharacteristic
34#14583094C2211yescharacteristic
35#14583094C2211yescharacteristic
36#14583051—21no—
37#14583045—21no—
38#14583032—21no—
39C4×#10621882C2211yescharacteristic
40#14582978C2211yescharacteristic
41D4×C43⋊C8C2211yescharacteristic
42D4×C43⋊C8C2211yescharacteristic, complemented
43C2×#13063258—21no—
44C2×#13063245—21no—
45#14582651C2211yescharacteristic, complemented
46#14582650C2211yescharacteristic, complemented
47#14582646C2211yescharacteristic, complemented
48#14582645C2211yescharacteristic, complemented
49#14582645C2211yescharacteristic, complemented
50#14582642C2211yescharacteristic
51D4⋊(C43⋊C8)C2211yescharacteristic
52D4⋊(C43⋊C8)C2211yescharacteristic, complemented
53D4⋊(C43⋊C8)C2211yescharacteristic, complemented
54#14582626C2211yescharacteristic
55#14582620C2211yescharacteristic, complemented
56C2×#135×C43—21no—
57C43×#547C2211yescharacteristic
58C43×#547C2211yescharacteristic, complemented
59C43×#494C2211yescharacteristic, complemented
60C43×#467—21no—
61C43×#461C2211yescharacteristic
62C43×#461C2211yescharacteristic, complemented
63C43×#452—21no—
64C43×#452—21no—
65C4×D8×C43C2211yescharacteristic
66C8×D4×C43C2211yescharacteristic, complemented
67C2×#10621900—21no—
68C22×D172—21no—
69C2×D4×D43—21no—
70#13063349—44no—
71C2×#10621873—21no—
72C2×#10621873—21no—
73C2×#10621883—21no—
74C2×#10621882—21no—
75C2×#10621882—21no—
76C2×#10621882D411yescharacteristic
77C2×#10621872—21no—
78C22×C43⋊C8—21no—
79C22×C43⋊C8—21no—
80#13063325C2311yescharacteristic
81#13063322C2311yescharacteristic
82#13063309—41no—
83#13063305—21no—
84#13063300—21no—
85#13063299—21no—
86#13063299—21no—
87#13063295—41no—
88C4×D172C2311yescharacteristic
89C2×C4⋊Dic43—21no—
90#13063258—21no—
91#13063258—21no—
92#13063258—21no—
93#13063258—21no—
94#13063258—21no—
95#13063258—41no—
96#13063258D411yescharacteristic
97C4⋊(C43⋊C8)C2311yescharacteristic
98C4⋊(C43⋊C8)C2311yescharacteristic
99C4⋊(C43⋊C8)C2311yescharacteristic
100#13063248—21no—
101#13063246—21no—
102#13063246D411yescharacteristic
103#13063245—21no—
104#13063245—21no—
105#13063245—21no—
106#13063245—21no—
107#13063245—21no—
108#13063245—21no—
109#13063245—41no—
110#13063245D411yescharacteristic
111C22⋊(C43⋊C8)—21no—
112C22⋊(C43⋊C8)—21no—
113(C2×D43)⋊C8—21no—
114C4×C43⋊C8C2311yescharacteristic
115C4×C43⋊C8C2311yescharacteristic
116C2×#41×C43—21no—
117C2×#41×C43—21no—
118#135×C43—44no—
119C2×Q16×C43—21no—
120C2×Q16×C43—21no—
121C2×SD16×C43—21no—
122C2×SD16×C43—21no—
123C2×D8×C43D411yescharacteristic
124C22×C8×C43—21no—
125C4⋊Q8×C43C2311yescharacteristic
126C4⋊Q8×C43C2311yescharacteristic
127#123×C43—41no—
128#123×C43—41no—
129C22⋊Q8×C43—21no—
130C22⋊Q8×C43—21no—
131#121×C43—21no—
132#121×C43—21no—
133C4×D4×C43C2311yescharacteristic
134C4×D4×C43C2311yescharacteristic
135C4×D4×C43C2311yescharacteristic
136C2×C4⋊C4×C43—21no—
137C2×C4⋊C4×C43—21no—
138#107×C43D411yescharacteristic
139C4⋊C8×C43C2311yescharacteristic
140#103×C43—21no—
141#103×C43—21no—
142#102×C43D411yescharacteristic
143#102×C43D411yescharacteristic
144C22⋊C8×C43—21no—
145C4×C8×C43C2311yescharacteristic
146C23×D43—41no—
147#41×C43—42no—
148#41×C43—42no—
149#41×C43—81no—
150#41×C43—81no—
151C2×Q8×C43—21no—
152C2×Q8×C43—21no—
153C2×D4×C43—41no—
154C2×D4×C43—41no—
155C2×D4×C43C2×D411yescharacteristic
156C2×D4×C43C2×D411yescharacteristic
157C2×D4×C43C2×D411yescharacteristic
158C22×C4×C43—21no—
159C22×C4×C43—21no—
160C22×C4×C43—21no—
161C22×C4×C43—41no—
162C22×C4×C43—41no—
163C2×C22⋊D43—41no—
164C2×C22⋊D43—41no—
165C22×Dic43—41no—
166#10621900—42no—
167#10621900—81no—
168D4×D43—42no—
169D4×D43—81no—
170C2×D172—21no—
171C2×D172—21no—
172C2×D172—21no—
173C2×D172—41no—
174C2×D172—42no—
175C2×D172—81no—
176C2×D172C2×D411yescharacteristic
177C2×C4×D43—21no—
178C2×Dic86—21no—
179Q16×C43—42no—
180Q16×C43—81no—
181SD16×C43—42no—
182SD16×C43—42no—
183D8×C43D814yescomplemented
184C2×C8×C43—21no—
185C2×C8×C43—42no—
186C2×C8×C43C2×D411yescharacteristic
187C2×C8×C43C2×D411yescharacteristic
188C4⋊C4×C43—21no—
189C4⋊C4×C43—21no—
190C4⋊C4×C43—21no—
191C4⋊C4×C43—21no—
192C4⋊C4×C43—41no—
193C4⋊C4×C43—41no—
194C4⋊C4×C43C2×D411yescharacteristic
195C4⋊C4×C43C2×D411yescharacteristic
196C4⋊C4×C43C2×D411yescharacteristic
197C22⋊C4×C43—21no—
198C22⋊C4×C43—21no—
199C22⋊C4×C43—21no—
200C22⋊C4×C43—41no—
201C22⋊C4×C43—41no—
202C42×C43C2411yescharacteristic
203#10621883—42no—
204#10621882—41no—
205#10621882—42no—
206#10621882—81no—
207(C2×D43)⋊C4—21no—
208(C2×D43)⋊C4—41no—
209(C2×D43)⋊C4—41no—
210C4⋊Dic43—21no—
211C4⋊Dic43—21no—
212C4⋊Dic43—41no—
213C4⋊Dic43C2×D411yescharacteristic
214C2×C43⋊C8—21no—
215C2×C43⋊C8—21no—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8—42no—
219C2×C43⋊C8—81no—
220C2×C43⋊C8C2×D411yescharacteristic
221C2×C43⋊C8C2×D411yescharacteristic
222#10621873—42no—
223#10621873—81no—
224#10621872—42no—
225C23×C43—21no—
226C23×C43—21no—
227C23×C43—21no—
228C22×D43—21no—
229C22×D43—41no—
230C22×D43—82no—
231C22×D43—82no—
232C22×D43—161no—
233Q8×C43—41no—
234Q8×C43—41no—
235Q8×C43—41no—
236Q8×C43—41no—
237D4×C43—21no—
238D4×C43—21no—
239D4×C43—21no—
240D4×C43—21no—
241D4×C43—82no—
242D4×C43—82no—
243D4×C43C2×D812yes—
244D4×C43C2×D812yes—
245C2×C4×C43—21no—
246C2×C4×C43—21no—
247C2×C4×C43—21no—
248C2×C4×C43—41no—
249C2×C4×C43—41no—
250C2×C4×C43—41no—
251C2×C4×C43—41no—
252C2×C4×C43—41no—
253C2×C4×C43—41no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—82no—
257C2×C4×C43—82no—
258C2×C4×C43C22×D411yescharacteristic
259C2×C4×C43C22×D411yescharacteristic
260C2×C4×C43H(5,𝔽2)11yescharacteristic
261C22⋊D43—82no—
262C22⋊D43—82no—
263C2×Dic43—21no—
264C2×Dic43—41no—
265C2×Dic43—82no—
266D172—21no—
267D172—21no—
268D172—41no—
269D172—41no—
270D172—41no—
271D172—41no—
272D172—81no—
273D172—82no—
274C4×D43—41no—
275C4×D43—41no—
276Dic86—41no—
277Dic86—41no—
278C8×C43—21no—
279C8×C43—41no—
280C8×C43C2×D812yes—
281C43⋊C8—21no—
282C43⋊C8—21no—
283C43⋊C8—41no—
284C43⋊C8—41no—
285C43⋊C8—41no—
286C43⋊C8—41no—
287#19758—431nomaximal, Sylow
288C22×C43—22no—
289C22×C43—22no—
290C22×C43—41no—
291C22×C43—41no—
292C22×C43—41no—
293C22×C43—41no—
294C22×C43—41no—
295C22×C43—41no—
296C22×C43—41no—
297C22×C43D4211yescharacteristic
298C2×D43—41no—
299C2×D43—41no—
300C2×D43—41no—
301C2×D43—81no—
302C2×D43—81no—
303C2×D43—81no—
304C2×D43—164no—
305C4×C43—21no—
306C4×C43—41no—
307C4×C43—41no—
308C4×C43—41no—
309C4×C43—81no—
310C4×C43—81no—
311C4×C43C22×D811yescharacteristic
312C4×C43#57511yescharacteristic
313C4×C43#57611yescharacteristic
314C4×C43#57711yescharacteristic
315Dic43—41no—
316Dic43—81no—
317#3334—431no—
318#3294—431no—
319#3283D4311yescharacteristic, complemented
320#3272—431no—
321#1722—431no—
322#1689—431no—
323#1681—431no—
324#1673—431no—
325#1665—431no—
326#1647—431no—
327#1647—431no—
328#1639—431no—
329#1625—431no—
330#1581—431no—
331#1580—431no—
332C2×C43—42no—
333C2×C43—42no—
334C2×C43—81no—
335C2×C43#326711yescharacteristic
336C2×C43D4×D811yescharacteristic
337C2×C43#328911yescharacteristic
338D43—81no—
339D43—161no—
340C2×AGL(1,ℤ/8)—861no—
341C2×#135—21no—
342#547—431no—
343#547C2×D4311yescharacteristic
344#547C2×D4311yescharacteristic, complemented
345D42—431no—
346#500—431no—
347#500—431no—
348#494C2×D4311yescharacteristic, complemented
349#492—431no—
350#486—431no—
351#486—431no—
352#480—861no—
353#480—861no—
354#480—861no—
355#474—431no—
356#474—431no—
357#474—431no—
358#468—861no—
359#467—21no—
360#466—861no—
361#466—861no—
362#461—431no—
363#461C2×D4311yescharacteristic
364#461C2×D4311yescharacteristic, complemented
365#459—431no—
366#459—431no—
367#452—21no—
368#452—21no—
369#450—861no—
370#449—861no—
371#447—861no—
372C4×D8—431no—
373C4×D8C2×D4311yescharacteristic
374#436—431no—
375C8×D4—431no—
376C8×D4—431no—
377C8×D4C2×D4311yescharacteristic, complemented
378C2×#107—861no—
379C2×#102—861no—
380#335—431no—
381#334—431no—
382#332—431no—
383#332—431no—
384#332—431no—
385#331—431no—
386D4⋊C8—431no—
387D4⋊C8—431no—
388D4⋊C8—431no—
389D4⋊C8—431no—
390#322—431no—
391C43#1975811yescharacteristic, complemented, Sylow
392C2×#41—21no—
393C2×#41—21no—
394C2×#41—861no—
395C22×D4—861no—
396C22×D4—861no—
397AGL(1,ℤ/8)—1724no—
398#135—44no—
399C2×Q16—21no—
400C2×Q16—21no—
401C2×SD16—21no—
402C2×SD16—21no—
403C2×SD16—861no—
404C2×SD16—861no—
405C2×SD16—861no—
406C2×D8—431no—
407C2×D8—861no—
408C2×D8—861no—
409C2×D8C22⋊D4311yescharacteristic
410C2×M16—861no—
411C22×C8—21no—
412C22×C8—861no—
413C22×C8—861no—
414C4⋊Q8—431no—
415C4⋊Q8C22×D4311yescharacteristic
416C4⋊Q8C22×D4311yescharacteristic
417#127—431no—
418#123—41no—
419#123—41no—
420#123—1721no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C22⋊Q8—861no—
424#121—21no—
425#121—21no—
426#121—861no—
427#121—861no—
428#121—861no—
429#120—1721no—
430C4×D4—431no—
431C4×D4C22×D4311yescharacteristic
432C4×D4C22×D4311yescharacteristic
433C4×D4C22×D4311yescharacteristic
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—861no—
437#107—431no—
438#107—861no—
439#107—861no—
440#107—861no—
441#107—861no—
442#107—861no—
443#107—1721no—
444#107C22⋊D4311yescharacteristic
445C4⋊C8—431no—
446C4⋊C8—431no—
447C4⋊C8—431no—
448C4⋊C8C22×D4311yescharacteristic
449#103—21no—
450#103—21no—
451#103—861no—
452#102—431no—
453#102—431no—
454#102—861no—
455#102—861no—
456#102—861no—
457#102—861no—
458#102—861no—
459#102—861no—
460#102—861no—
461#102—1721no—
462#102C22⋊D4311yescharacteristic
463#102C22⋊D4311yescharacteristic
464C22⋊C8—21no—
465C22⋊C8—861no—
466C22⋊C8—861no—
467C22⋊C8—861no—
468C4×C8—431no—
469C4×C8—431no—
470C4×C8C22×D4311yescharacteristic
471C24—1721no—
472#41—42no—
473#41—42no—
474#41—81no—
475#41—81no—
476#41—3441no—
477#41—1722no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—861no—
481C2×D4—41no—
482C2×D4—41no—
483C2×D4—431no—
484C2×D4—861no—
485C2×D4—861no—
486C2×D4—861no—
487C2×D4—3441no—
488C2×D4—3441no—
489C2×D4—1721no—
490C2×D4—1722no—
491C2×D4—1721no—
492C2×D4—1721no—
493C2×D4—1722no—
494C2×D4D4×D4311yescharacteristic
495C2×D4C2×C22⋊D4311yescharacteristic
496C2×D4C2×C22⋊D4311yescharacteristic
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—861no—
503C22×C4—1721no—
504Q16—42no—
505Q16—81no—
506SD16—42no—
507SD16—42no—
508SD16—3441no—
509SD16—1722no—
510SD16—1722no—
511D8—3441no—
512D8—1721no—
513D8—1722no—
514D8#1062188214yescomplemented
515M16—1722no—
516C2×C8—21no—
517C2×C8—42no—
518C2×C8—431no—
519C2×C8—431no—
520C2×C8—861no—
521C2×C8—861no—
522C2×C8—861no—
523C2×C8—861no—
524C2×C8—3441no—
525C2×C8—1722no—
526C2×C8C2×C22⋊D4311yescharacteristic
527C2×C8C2×C22⋊D4311yescharacteristic
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—431no—
535C4⋊C4—861no—
536C4⋊C4—861no—
537C4⋊C4—1721no—
538C4⋊C4D4×D4311yescharacteristic
539C4⋊C4C2×C22⋊D4311yescharacteristic
540C4⋊C4C2×C22⋊D4311yescharacteristic
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—861no—
547C22⋊C4—1721no—
548C22⋊C4—1721no—
549C42C23×D4311yescharacteristic
550C23—21no—
551C23—21no—
552C23—21no—
553C23—861no—
554C23—3442no—
555C23—3442no—
556C23—1721no—
557C23—6881no—
558Q8—41no—
559Q8—41no—
560Q8—41no—
561Q8—41no—
562Q8—1721no—
563Q8—1721no—
564D4—21no—
565D4—21no—
566D4—21no—
567D4—21no—
568D4—82no—
569D4—82no—
570D4—861no—
571D4—861no—
572D4—3441no—
573D4—3442no—
574D4—3442no—
575D4—3442no—
576D4—1721no—
577D4—1721no—
578D4—1721no—
579D4—1721no—
580D4C2×#1062188212yes—
581D4C2×#1062188212yes—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—861no—
596C2×C4—3442no—
597C2×C4—1721no—
598C2×C4—1721no—
599C2×C4—1721no—
600C2×C4C2×D4×D4311yescharacteristic
601C2×C4C22×C22⋊D4311yescharacteristic
602C2×C4#1306337011yescharacteristic
603C8—21no—
604C8—41no—
605C8—861no—
606C8—861no—
607C8—1721no—
608C8—1721no—
609C8—1721no—
610C8—1721no—
611C8C2×#1062188212yes—
612C22—22no—
613C22—22no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—3441no—
622C22—3441no—
623C22—3441no—
624C22—1721no—
625C22—1721no—
626C22—1721no—
627C22—6884no—
628C22D4×C22⋊D4311yescharacteristic
629C4—21no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—3441no—
636C4—1721no—
637C4C22×#1062188211yescharacteristic
638C4#1458369611yescharacteristic
639C4#1458370211yescharacteristic
640C4#1458370611yescharacteristic
641C2—42no—
642C2—42no—
643C2—81no—
644C2—3441no—
645C2—6881no—
646C2#1914612011yescharacteristic
647C2D4×#1062188211yescharacteristic
648C2#1914636611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D43
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3283C2
socleC22×C43D42
hypercenter#3283D43
hypocenterC43#19758
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ #3283

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #19146938C2semidirect
2. #19146402C2semidirect
3. #19146196C2semidirect
4. #19134267C2semidirect
5. #19134081C2semidirect
6. #19134034C2semidirect
7. #19133982C2semidirect
8. #19133936C2semidirect
9. #19133830C2semidirect
10. #19133828C2semidirect
11. #19133772C2semidirect
12. #19133684C2semidirect
13. #19133424C2semidirect
14. #19133419C2semidirect
15. C43×#3283C2semidirect
19. #14583302C22semidirect
20. #14583302C22semidirect
27. #14583179C22semidirect
42. D4×C43⋊C8C22semidirect
45. #14582651C22semidirect
46. #14582650C22semidirect
47. #14582646C22semidirect
48. #14582645C22semidirect
49. #14582645C22semidirect
52. D4⋊(C43⋊C8)C22semidirect
53. D4⋊(C43⋊C8)C22semidirect
55. #14582620C22semidirect
58. C43×#547C22semidirect
59. C43×#494C22semidirect
62. C43×#461C22semidirect
66. C8×D4×C43C22semidirect
183. D8×C43D8semidirect
319. #3283D43semidirect
344. #547C2×D43semidirect
348. #494C2×D43semidirect
364. #461C2×D43semidirect
377. C8×D4C2×D43semidirect
391. C43#19758semidirect
514. D8#10621882semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#191469382
#191464022
#191461962
#191342672
#191340812
#191340342
#191339822
#191339362
#191338302
#191338282
#191337722
#191336842
#191334242
#191334192
C43×#32832
#1975843